In linear algebra, a square matrix with complex entries is said to be skew-Hermitian or antihermitian if its conjugate transpose is the negative of the original matrix. That is, the matrix is skew-Hermitian if it satisfies the relation
where denotes the conjugate transpose of the matrix . In component form, this means that
for all indices and , where is the element in the -th row and -th column of , and the overline denotes complex conjugation.
Skew-Hermitian matrices can be understood as the complex versions of real skew-symmetric matrices, or as the matrix analogue of the purely imaginary numbers. The set of all skew-Hermitian matrices forms the Lie algebra, which corresponds to the Lie group U(n). The concept can be generalized to include linear transformations of any complex vector space with a sesquilinear norm.
Note that the adjoint of an operator depends on the scalar product considered on the dimensional complex or real space . If denotes the scalar product on , then saying is skew-adjoint means that for all one has
Imaginary numbers can be thought of as skew-adjoint (since they are like matrices), whereas real numbers correspond to self-adjoint operators.
For example, the following matrix is skew-Hermitian
- The eigenvalues of a skew-Hermitian matrix are all purely imaginary (and possibly zero). Furthermore, skew-Hermitian matrices are normal. Hence they are diagonalizable and their eigenvectors for distinct eigenvalues must be orthogonal.
- All entries on the main diagonal of a skew-Hermitian matrix have to be pure imaginary; i.e., on the imaginary axis (the number zero is also considered purely imaginary).
- If and are skew-Hermitian, then is skew-Hermitian for all real scalars and .
- is skew-Hermitian if and only if (or equivalently, ) is Hermitian.
- is skew-Hermitian if and only if the real part is skew-symmetric and the imaginary part is symmetric.
- If is skew-Hermitian, then is Hermitian if is an even integer and skew-Hermitian if is an odd integer.
- is skew-Hermitian if and only if for all vectors .
- If is skew-Hermitian, then the matrix exponential is unitary.
- The space of skew-Hermitian matrices forms the Lie algebra of the Lie group .
Decomposition into Hermitian and skew-Hermitian
- The sum of a square matrix and its conjugate transpose is Hermitian.
- The difference of a square matrix and its conjugate transpose is skew-Hermitian. This implies that the commutator of two Hermitian matrices is skew-Hermitian.
- An arbitrary square matrix can be written as the sum of a Hermitian matrix and a skew-Hermitian matrix :